225 ideas
| 20866 | Wise men participate in politics, especially if it shows moral progress [Stoic school, by Stobaeus] |
| 20854 | Wise men are never astonished at things which other people take to be wonders [Stoic school, by Diog. Laertius] |
| 20815 | No wise man has yet been discovered [Stoic school, by Cicero] |
| 20806 | Stoic physics concerns cosmos, elements and causes (with six detailed divisions) [Stoic school, by Diog. Laertius] |
| 20839 | Ethics studies impulse, good, passion, virtue, goals, value, action, appropriateness, encouragement [Stoic school, by Diog. Laertius] |
| 20867 | True philosophising is not memorising ideas, but living by them [Stoic school, by Stobaeus] |
| 21675 | Some facts are indispensable for an effect, and others actually necessitate the effect [Stoic school, by Cicero] |
| 21810 | The Stoics distinguished spoken logos from logos within the mind [Stoic school, by Plotinus] |
| 20775 | Stoics study canons, criteria and definitions, in order to find the truth [Stoic school, by Diog. Laertius] |
| 21393 | Stoics believed that rational capacity in man (logos) is embodied in the universe [Stoic school, by Long] |
| 20776 | Dialectics is mastery of question and answer form [Stoic school, by Diog. Laertius] |
| 18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
| 20849 | Falsehoods corrupt a mind, producing passions and instability [Stoic school, by Diog. Laertius] |
| 20823 | The truth bearers are said to be the signified, or the signifier, or the meaning of the signifier [Stoic school, by Sext.Empiricus] |
| 20778 | Stoics like syllogisms, for showing what is demonstrative, which corrects opinions [Stoic school, by Diog. Laertius] |
| 13439 | Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock] |
| 13421 | 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock] |
| 13422 | 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock] |
| 13355 | 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock] |
| 13350 | 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock] |
| 13351 | 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock] |
| 13356 | The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock] |
| 13352 | 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock] |
| 13353 | 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock] |
| 13354 | 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock] |
| 13610 | A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock] |
| 18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
| 13846 | A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock] |
| 18114 | There is no single agreed structure for set theory [Bostock] |
| 18107 | A 'proper class' cannot be a member of anything [Bostock] |
| 18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
| 18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
| 18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| 13346 | Truth is the basic notion in classical logic [Bostock] |
| 13545 | Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock] |
| 13822 | Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock] |
| 13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
| 13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
| 13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
| 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13799 | The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock] |
| 13800 | |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock] |
| 13803 | If we are to express that there at least two things, we need identity [Bostock] |
| 21400 | Stoics avoided universals by paraphrasing 'Man is...' as 'If something is a man, then it is...' [Stoic school, by Long] |
| 13357 | Truth-functors are usually held to be defined by their truth-tables [Bostock] |
| 20788 | The contradictory of a contradictory is an affirmation [Stoic school, by Diog. Laertius] |
| 13812 | A 'zero-place' function just has a single value, so it is a name [Bostock] |
| 13811 | A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock] |
| 13360 | In logic, a name is just any expression which refers to a particular single object [Bostock] |
| 13361 | An expression is only a name if it succeeds in referring to a real object [Bostock] |
| 13814 | Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock] |
| 13816 | Because of scope problems, definite descriptions are best treated as quantifiers [Bostock] |
| 13817 | Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock] |
| 13848 | We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock] |
| 13813 | Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock] |
| 13815 | Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock] |
| 13438 | 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock] |
| 13818 | If we allow empty domains, we must allow empty names [Bostock] |
| 18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
| 13801 | An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock] |
| 13619 | Quantification adds two axiom-schemas and a new rule [Bostock] |
| 13622 | Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock] |
| 13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
| 13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
| 13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
| 13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
| 18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
| 13753 | Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock] |
| 13755 | Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock] |
| 13758 | In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock] |
| 13754 | Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock] |
| 13611 | Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock] |
| 13613 | A completed open branch gives an interpretation which verifies those formulae [Bostock] |
| 13612 | Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock] |
| 13761 | In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock] |
| 13757 | Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock] |
| 13756 | A tree proof becomes too broad if its only rule is Modus Ponens [Bostock] |
| 13762 | Tableau rules are all elimination rules, gradually shortening formulae [Bostock] |
| 13759 | Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock] |
| 13760 | A sequent calculus is good for comparing proof systems [Bostock] |
| 13364 | Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG] |
| 13821 | Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock] |
| 13362 | If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock] |
| 13541 | For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock] |
| 13542 | A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock] |
| 13540 | A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock] |
| 13544 | Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock] |
| 13618 | Compactness means an infinity of sequents on the left will add nothing new [Bostock] |
| 18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
| 18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
| 18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
| 18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
| 18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
| 18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
| 18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
| 18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
| 18097 | The Peano Axioms describe a unique structure [Bostock] |
| 13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
| 13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock] |
| 18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
| 18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
| 18149 | There are many criteria for the identity of numbers [Bostock] |
| 18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
| 18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
| 18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
| 18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
| 18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
| 18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| 18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
| 18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
| 18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
| 18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
| 18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
| 18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
| 18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
| 18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
| 18140 | The best version of conceptualism is predicativism [Bostock] |
| 18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
| 18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
| 18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
| 18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
| 18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
| 18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
| 18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
| 21594 | Stoics applied bivalence to sorites situations, so everyone is either vicious or wholly virtuous [Stoic school, by Williamson] |
| 20824 | Stoics have four primary categories: substrates, qualities, dispositions, relative dispositions [Stoic school, by Simplicius] |
| 13802 | Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock] |
| 13543 | A relation is not reflexive, just because it is transitive and symmetrical [Bostock] |
| 14329 | Some dispositional properties (such as mental ones) may have no categorical base [Price,HH] |
| 20817 | Platonic Forms are just our thoughts [Stoic school, by Ps-Plutarch] |
| 6037 | Stoics say matter has qualities, and substance underlies it, with no form or qualities [Stoic school, by Chalcidius] |
| 20826 | How is separateness possible, if separated things are always said to be united? [Alexander on Stoic school] |
| 20825 | How is divisibility possible, if stoics say things remain united when they are divided? [Alexander on Stoic school] |
| 20872 | Stoics say wholes are more than parts, but entirely consist of parts [Stoic school, by Sext.Empiricus] |
| 13847 | If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock] |
| 13820 | The idea that anything which can be proved is necessary has a problem with empty names [Bostock] |
| 20790 | A proposition is possible if it is true when nothing stops it being true [Stoic school, by Diog. Laertius] |
| 20789 | Conditionals are false if the falsehood of the conclusion does not conflict with the antecedent [Stoic school, by Diog. Laertius] |
| 20783 | Knowledge is a secure grasp of presentations which cannot be reversed by argument [Stoic school, by Diog. Laertius] |
| 20868 | Two sorts of opinion: either poorly grounded belief, or weak belief [Stoic school, by Stobaeus] |
| 20784 | There are non-sensible presentations, which come to us through the intellect [Stoic school, by Diog. Laertius] |
| 20803 | Stoics say we are born like a blank sheet of paper; the first concepts on it are sensations [Stoic school, by Ps-Plutarch] |
| 6025 | At birth the soul is a blank sheet ready to be written on [Stoic school, by Aetius] |
| 20781 | Non-graspable presentations are from what doesn't exist, or are not clear and distinct [Stoic school, by Diog. Laertius] |
| 20792 | Stoic perception is a presentation to which one voluntarily assents [Stoic school, by Stobaeus] |
| 20805 | All our concepts come from experience, directly, or by expansion, reduction or compounding [Stoic school, by Sext.Empiricus] |
| 20782 | Dialectic is a virtue which contains other virtues [Stoic school, by Diog. Laertius] |
| 1772 | For Stoics knowledge is an assertion which never deviates from the truth [Stoic school, by Diog. Laertius] |
| 20779 | Demonstration derives what is less clear from what is clear [Stoic school, by Diog. Laertius] |
| 23251 | The Stoics think that soul in the narrow sense is nothing but reason [Stoic school, by Frede,M] |
| 20809 | Eight parts of the soul: five senses, seeds, speech and reason [Stoic school, by Diog. Laertius] |
| 23321 | Division of the soul divides a person, reducing responsibility for the nonrational part [Stoic school, by Frede,M] |
| 23267 | Stoics say the soul is a mixture of air and fire [Stoic school, by Galen] |
| 20785 | Our conceptions arise from experience, similarity, analogy, transposition, composition and opposition [Stoic school, by Diog. Laertius] |
| 9032 | Before we can abstract from an instance of violet, we must first recognise it [Price,HH] |
| 9035 | If judgement of a characteristic is possible, that part of abstraction must be complete [Price,HH] |
| 9034 | There may be degrees of abstraction which allow recognition by signs, without full concepts [Price,HH] |
| 9036 | There is pre-verbal sign-based abstraction, as when ice actually looks cold [Price,HH] |
| 9037 | Intelligent behaviour, even in animals, has something abstract about it [Price,HH] |
| 7502 | For Stoics the true self is defined by what I can be master of [Stoic school, by Foucault] |
| 23327 | Stoics expanded the idea of compulsion, and contracted what counts as one's own actions [Stoic school, by Frede,M] |
| 7672 | The free will problem was invented by the Stoics [Stoic school, by Berlin] |
| 23315 | The nearest to ancient determinism is Stoic fate, but that is controlled by a sympathetic God [Stoic school, by Frede,M] |
| 9033 | Recognition must precede the acquisition of basic concepts, so it is the fundamental intellectual process [Price,HH] |
| 4014 | Stoics classify passions according to the opinion of good and bad which they imply [Stoic school, by Taylor,C] |
| 23988 | There are four basic emotions: pleasure or delight, distress, appetite, and fear [Stoic school, by Cicero] |
| 6594 | Stoics said that correct judgement needs an invincible criterion of truth [Stoic school, by Fogelin] |
| 20804 | Concepts are intellectual phantasms [Stoic school, by Ps-Plutarch] |
| 10645 | We reach concepts by clarification, or by definition, or by habitual experience [Price,HH] |
| 9030 | Abstractions can be interpreted dispositionally, as the ability to recognise or imagine an item [Price,HH] |
| 9029 | If ideas have to be images, then abstract ideas become a paradoxical problem [Price,HH] |
| 10644 | A 'felt familiarity' with universals is more primitive than abstraction [Price,HH] |
| 10646 | Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances [Price,HH] |
| 9031 | The basic concepts of conceptual cognition are acquired by direct abstraction from instances [Price,HH] |
| 20786 | Predicates are incomplete 'lekta' [Stoic school, by Diog. Laertius] |
| 13363 | A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock] |
| 23322 | Humans have rational impressions, which are conceptual, and are true or false [Stoic school, by Frede,M] |
| 20777 | Rhetoric has three types, four modes, and four sections [Stoic school, by Diog. Laertius] |
| 18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
| 23323 | Earlier Stoics speak of assent, but not of choice, let alone of a will [Stoic school, by Frede,M] |
| 23305 | Stoics said responsibility depends on rationality [Stoic school, by Sorabji] |
| 1907 | Stoics use 'kalon' (beautiful) as a synonym for 'agathon' (good) [Bury on Stoic school] |
| 22757 | Stoics say that folly alone is evil [Stoic school, by Sext.Empiricus] |
| 20846 | Prime values apply to the life in agreement; useful values apply to the natural life [Stoic school, by Diog. Laertius] |
| 20847 | The appraiser's value is what is set by someone experienced in the facts [Stoic school, by Diog. Laertius] |
| 20870 | The goal is to live consistently with the constitution of a human being [Stoic school, by Clement] |
| 22238 | Stoics said health is an 'indifferent', but they still considered it preferable [Stoic school, by Pormann] |
| 20861 | The health of the soul is a good blend of beliefs [Stoic school, by Stobaeus] |
| 3553 | Stoic morality says that one's own happiness will lead to impartiality [Stoic school, by Annas] |
| 20851 | Virtuous men do not feel sexual desire, which merely focuses on physical beauty [Stoic school, by Diog. Laertius] |
| 7499 | Stoicism was an elitist option to lead a beautiful life [Stoic school, by Foucault] |
| 20843 | Final goods: confidence, prudence, freedom, enjoyment and no pain, good spirits, virtue [Stoic school, by Diog. Laertius] |
| 22753 | Happiness for the Stoics was an equable flow of life [Stoic school, by Sext.Empiricus] |
| 20865 | Happiness is the end and goal, achieved by living virtuously, in agreement, and according to nature [Stoic school, by Stobaeus] |
| 20840 | Stoics say pleasure is at most a byproduct of finding what is suitable for us [Stoic school, by Diog. Laertius] |
| 20852 | Rapture is a breakdown of virtue [Stoic school, by Diog. Laertius] |
| 6895 | If humans are citizens of the world (not just a state) then virtue is all good human habits [Stoic school, by Mautner] |
| 20848 | An appropriate action is one that can be defended, perhaps by its consistency. [Stoic school, by Diog. Laertius] |
| 20844 | Honour is just, courageous, orderly or knowledgeable. It is praiseworthy, or functions well [Stoic school, by Diog. Laertius] |
| 4012 | The Stoics rejected entirely the high value that had been placed on contemplation [Stoic school, by Taylor,C] |
| 5073 | Stoics do not despise external goods, but subject them to reason, and not to desire [Taylor,R on Stoic school] |
| 20862 | Crafts like music and letters are virtuous conditions, and they accord with virtue [Stoic school, by Stobaeus] |
| 5072 | For Stoics, obligations are determined by social role [Taylor,R on Stoic school] |
| 21396 | Man is distinguished by knowing conditional truths, because impressions are connected [Stoic school, by Long] |
| 1781 | Stoics favour a mixture of democracy, monarchy and aristocracy [Stoic school, by Diog. Laertius] |
| 21384 | The Stoics saw the whole world as a city [Stoic school, by Long] |
| 20859 | The best government blends democracy, monarchy and aristocracy [Stoic school, by Diog. Laertius] |
| 3561 | Stoics originated the concept of natural law, as agreed correct reasoning [Stoic school, by Annas] |
| 3046 | Stoics say a wise man will commit suicide if he has a good enough reason [Stoic school, by Diog. Laertius] |
| 20858 | Suicide is reasonable, for one's country or friends, or because of very bad health [Stoic school, by Diog. Laertius] |
| 3556 | Stoic 'nature' is deterministic, physical and teleological [Stoic school, by Annas] |
| 22743 | Unlike Epicurus, Stoics distinguish the Whole from the All, with the latter including the void [Stoic school, by Sext.Empiricus] |
| 13296 | The cosmos has two elements - passive matter, and active cause (or reason) which shapes it [Stoic school, by Seneca] |
| 20827 | The cosmos is regularly consumed and reorganised by the primary fire [Stoic school, by Aristocles] |
| 7815 | Early Stoics called the logos 'god', meaning not a being, but the principle of the universe [Stoic school] |
| 6038 | Stoics say god is matter, or an inseparable quality of it, or is the power within it [Stoic school, by Chalcidius] |
| 20829 | Virtuous souls endure till the end, foolish souls for a short time, animal souls not at all [Stoic school, by Eusebius] |
| 6039 | Stoics say virtuous souls last till everything ends in fire, but foolish ones fade away [Stoic school, by ] |